Here is the full list of all 37 The Slimeking's Tower achievements.
Unlock Hairry's secret item
Mind Completed
Mind Unlocked
Prayer Completed
Dots Completed
Prayer Unlocked
Dots Unlocked
Void Completed
Midas Completed
The Collector Completed
Void Unlocked
Jade Completed
The Collector Unlocked
Igneous Completed
Maiyan Completed
Hairry Completed
Beat the portal
Midas Unlocked
Igneous Unlocked
Reach The Portal
Reach Sigma
Reach Theta
Reach Tau
Reach Ohm
Reach Pi
Reach the basement
Jade Unlocked
Reach Omega
Attack a mimic before it's transformed
Defeat Lodeslime
Reach Gamma
Reach Epsilon
Reach Delta
Maiyan Unlocked
Reach Beta
Get your first boss key
Play Adventure Mode for the first time